ξ ε ξ ε Dε

CONDITIONS FOR THE CONVERGENCE OF BRANCHING PROCESSES WITH IMMIGRATION STARTING FROM A LARGE NUMBER OF PARTICLES

H. K. Zhumakulov

Ph. D., Associate Professor of Kokand State Pedagogical Institute A. Ergashev

Kokand SPI ANNOTATION

In this section, we study sufficient conditions for the convergence of a sequence of almost critical Galton -Watson branching processes with uniform immigration starting from a large number of particles.

Keywords ; Galton -Watson, branching processes , almost critical, random process, Skorokhod spaces

Let for everyone

n  N  

^{k j( ),n}

, , k j  N  and  

^{k( )n}

, k  N  are two independent sets of independent, non-negative, integer- valued and identically distributed random variables. For each n  N, we define the process by the  X

^{k( )n}

, k  N

⁰



following recursive relations

( )

( ) ( ) 1 ( ) ( )

0 ,

1

0, , .

n

Xk

n n n n

k k j k

j

X X

⁻

  k

=

= =  +  N

^(one)

If we interpret the value



_{k j}^{( )}_,ⁿ as the number of descendants of the j -th particle in the -th

k − 1

generation , and the value



_k^{( )n} - as the number of particles immigrating into the population in the k -th generation, then the value

X

_k^{( )n} is the number of particles in the population in the k - th generation. Due to this interpretation, process (1) is called the Galton -Watson branching process with immigration.

Let us assume that the values

( ) ( ) 2 ( )

1,1ⁿ

,

1ⁿ

,

1,1ⁿ

n n n

m = E   = E   = D 

^and

b

_n²

= D 

₁^{( )n}

are finite for everyone

n  N

. Process (1) is called subcritical , critical and supercritical if

1, 1, 1

n n n

m  m = m 

, respectively. If

m

_n

→ 1

^for

n → 

, then the sequence (1) is called almost critical.

Let , for each , be

n  N 

₀^{( )n}

−

a positive integer random variable,

{ 

_{k j}^{( )}_,ⁿ

, , k j  N }

^and

{ 

_k( )ⁿ

, k  N }

be two mutually independent collections of identically distributed, non-negative, integer- valued random variables that do not depend on



₀^{( )n .}

For each

n  N

^process

 Xk( )n , k  N0, we define it as follows:

( )

( ) ( ) ( ) 1 ( ) ( )

0 0 ,

1

, , .

n

Xk

n n n n n

k k j k

j

X  X

⁻

  k

=

= =  +  N

assume the existence of second moments of the quantities

X

₀^{( )n}

, 

_{k j}^{( )}_,ⁿ ,



_k^{( )n} and keep the same notation for the means and variances of the quantities



_{k j}^{( )}_,^{n and}



_k^{( )n} as at the beginning of § Next, we define a step process

X t

_n

( )

as an element of the Skorokhod space

D [0, ] T

^{, setting}

( ) [ ]

( )

ⁿ

, 0

n nt

X t = X t 

, where [a] means the integer part of the number a. In what follows, the sign

⎯⎯→

^D will mean weak convergence in the Skorokhod topology (see [1]).

The following theorem gives an idea of the asymptotic behavior of the process

X t t

_n

( ),  0

^for

n → 

in the case when

X

₀^{( )n}

⎯⎯

^

→

(at the beginning there are "many" particles) and

n

1

m →

^for

n → 

(almost the critical case).

Theorem . Let

1

n

1

n n

m o

d d

  

= + +  

 

^{, where}

  R

^and

d

_nbe some sequence of positive numbers such that



_n

= nd

_n⁻¹

→   

for

n → 

^{. Let}

  0

the following conditions be satisfied for some:

n

^1−



_n

→ 

and

n

^1−

b

_n²

→ b

²

, n

^1−



_n²

→ 0

,

n

^−

X

₀^{( )n}

⎯⎯

^P

→ X

₀^for

n → 

^,

lim

₀^{( )n}

n

n EX

^−

→

 

^and

EX

₀

 

. Then for any

T  0

( )

^D

( ), [0, ]

X t

n

t t T

n

^

⎯⎯→  

^at

n → 

^,

where the limiting random process

 ( ) t

has the following form

0

0

( 1), если 0,

( )

, если 0.

t t

X e e

t

X t









 

 

 + − 

=  

 + =



Proof of the theorem. We represent the value

X

_k^{( )n}₊₁in the form

( ) ( ) ( ) ( )

1

( 1) ,

n n n n

k k n k n k

X

₊

= X + m − X +  + M

(2)

where

( )

( ) ( ) ( )

, 1

1

( )

n

Xk

n n n

k k j n k n

j

M  m 

₊



=

=  − + −

. Obviously, this

M

_k^{( )n}

, k  N

₀forms a square- integrable martingale difference with respect to the flow of σ-algebras

F

_k^{( )n}

( ) ( ) ( )

0 1

{ X

ⁿ

, X

ⁿ

, , X

_kⁿ

}



=

^{. Let} _nk

X

^{k( )n}

n

^

 =

. Then from (2) we have

( )

1 ( )

0

0 1

1 1

, ( ( 1) )

n

n

n nk nk n nk n k

X n m n M

n

n n



 

 = 

₊

=  + −  + 

⁻

 +

⁽³⁾

It is easy to verify that

( ) ( )

0

1 1

k

n k n n

k n n

n

EX m EX m

m − 

= +

−

^{. (four)}

Further, applying Doob's inequality for martingales , we have

[ ] [ ] 2

( ) 2 2 ( )

0 0

0

sup 1

nt nT

n n

n k k

k k

t T

I P M n E M

n





 

^{− −}

= =

 

   

=             =

[ ]

2 2 2 ( ) 2

0

[ ]

nT n

n k n

k

n

^

EX nT b



^{− −}



=

 

=  + 

  

for any

  0

. From here and from (4), after simple calculations, we obtain the estimate

[ ] 1 2 [ ] 1

2 2 2 ( ) 2

0

1 1

[ ] [ ]

1 1 1

nT nT

n n n n n

n n n

n n n

m m

I n EX nT nT b

m m m



 



^{− −}

 

⁺

− 

⁺

−  

    − + −   − −   +   

^{. (5)}

Obviously, if

  0

^{, then}

n

m

n

→ e

^at

n → 

^.

Then from (5) we have

( )

2 1 1 2 1 0

( ( 1)

n T

n n n

I n e E X

n

 

 

^{− − −}

 

⁻ 

 − +

1 1 2 1 1

( (

^T

1) )

n n n n

n

^−

 n

^−

  

^{− −}

e

^

T

+ − − +

1 2 2

) (1).

n

^

b T

n

o



⁻

+ +

If

 = 0

^{, then}

( )

n

1

n n n

m = +  + o 

and from (5) it follows that

( )

2 1 2 0 1 1 2 1 2 2

n

n n n n n

I n E X n n n b

n

   



⁻



⁻



 ⁻



⁻



⁻



  + + 

 

^.

Then, taking into account the conditions of the theorem, we obtain that

n

0

I →

^at

n → 

^.

Now, applying Theorem 3.1 from [2] and taking into account the last relations, we obtain

1

max

nk nk ^P

0

k nT

 Z

 

− ⎯⎯ →

^at

n → 

^{, (6)}

( )

1 1

0

0 1

, ( ) 1

n

n nk nk n nk n

Z X Z Z d Z n

n n



 ₊



⁻



⁻

= = + + 

^.

Further, applying Theorem 3.2 from [2], we have

0 1

( ) ( ) 0

sup

n

max

nk ^P

t T k nT

Z t t Z k

  n

   

− = −       ⎯⎯ →

for

n → 

^{, where}

Z t

_n

( ) = Z

_{n nt[ ]}, the process

 ( ) t

is a solution of the differential equation

( ) ( ( ))

d  t =   + t dt

with initial condition

 (0) = X

₀. From here and (6) it follows that

1 1

0

( ) ( ) 0

sup

ⁿ

max

nk nk

max

nk ^P

k nT k nT

t T

X t k

t Z Z

n

^

   n

   

 

−  − − −       ⎯⎯ →

for

n → 

, which was to be proved. The proof of the theorem is complete.

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Vol.3. –496 S.

2.

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6.

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